We provide a simple construction of
a function F:**R**^{2}-->**R** discontinuous on a perfect set P, while
having continuous restrictions F|C for all twice differentiable curves C.
In particular, F is separately continuous and linearly continuous.
While it has been known that the projection \pi[P] of any such set P
onto a straight line must be meager, our construction allows
\pi[P] to have arbitrarily large measure.
In particular, P can
have arbitrarily large 1-Hausdorff measure,
which is the best possible result in this direction, since any such P has Hausdorff
dimension at most 1.

**Full text on line in pdf format**.

**Last modified September 27, 2012.**